The Unapologetic Mathematician. John Armstrong's self-described "blath", The Unapologetic Mathematician: Mathematics for the Interested Outsider, is a great place to read almost-daily installments on higher level mathematics. After many months of posts on representation theory and symmetric groups, recent entries focus on differential topology by introducing manifolds. Interested graduate students or anyone with some basic point set topology knowledge can appreciate posts concerning different types of manifolds, their dimensions, and other basic properties. Although the sentiment of the subtitle (that laymen can follow the main lines of exposition) is overly optimistic, the continuity from post to post sets this blog apart. In addition, internal links to previous posts make unfamiliar terms easy to look up. Dr. Armstrong started his blog in 2007 after earning his Ph.D. from Yale studying category theory. He now earns his living by programming and has begun a new blog, The Unapologetic Programmer--Adventures of a Former Mathematician.

Fano Varieties and Extremal Laurent Polynomials: A collaborative research blog. Beautiful pictures and animations reminiscent of flowers or folding cloth grace recent posts of this blog, in which most entries take the form of a technical conversation between specialists Tom Coates, Alessio Corti, Sergei Galkin, Vasily Golyshev, and Al Kasprzyk. In their post on February 9th, the collaborators take a break from the language of research and describe the images of surfaces within a three dimensional Fano variety as being analogous to the MRI images made to help map the brain.

Because the main purpose of the blog is to enable researchers peppered across the globe to collaborate, exclusivity is to be expected. Nonetheless, readers who peruse other parts of the blog will appreciate titles of posts, such as "Things are not as straightforward as they seem", that give some insight into the trials and victories of mathematical research. And by playing the short animations, we may all appreciate at least some of the fruits of these mathematical labors.

Tom Coates's research is also covered in "Atoms Ripple in the Periodic Table of Shapes," by Jacob Aron (New Scientist, 16 February 2011). Here Coates couches the research in terms that a layperson might appreciate by describing part of the groups research project as an effort to create a "periodic table" of shapes. See more in this month's Math Digest: "A new dimension for mathematics--the Periodic Table of shapes,".

Where can you play with a ring of fire, hold a Menger Sponge, tile the floor with monkeys, and buy a Museum of Math t-shirt? Nowhere yet - the Museum of Math opens in 2012 - but Wall Street Journal columnist and self-proclaimed math-hater Robert Gardner got a preview of these attractions while chatting with MOMATH founder Glen Whitney. In his column about the museum, Gardner acknowledges the appeal of mathematical discovery, while wondering how it can be conveyed in an interactive museum. Whitney, a former hedge-fund algorithm manager with Renaissance Technologies, tries to answer this question with exhibits like "The Ring of Fire," a laser-lit circle which reveals the geometric structures in a clear plastic cylinder. The museum will start out in a small, rented space, and the trick, according to Whitney, will be not filling the space but rather deciding what to leave out. Whitney continues to raise funds for the museum, and is in negotiations for street-front space. Even without a space, MOMATH began a monthly public lecture series called Math Encounters on March 3, with Erik Demaine's talk "The Geometry of Origami."

The periodic table of elements may soon be joined by a periodic table of varieties, although it's more likely to sit in volumes on your bookshelf than on a poster on your wall. The first article--appearing on the science and tech news website Gizmag--reports on the initiation of a three-year project, headed by algebraic geometers Tom Coates and Alessio Corti (both of Imperial College London), to classify the basic building blocks of three-, four-, and five-dimensional shapes. A modeling program developed by Coates will isolate shapes which are indivisible in some algebro-geometric sense. The researchers—Corti, Coates, Vasily Golyshev (IITP, Moscow), Al Kasprzyk (Imperial College London), Sergey Galkin (IPMU, Tokyo), and the Magma team (the Computational Algebra Group, University of Sydney, Australia)--then hope to understand the properties of these shape "atoms" from the algebraic equations defining them, and ultimately to understand how their properties affect those of shapes "built" from them. "We want to build a theory of chemistry for shapes," the article quotes Dr. Coates as saying. The item in Science notes that the team is searching for a few thousand basic building blocks among almost 500 million four-dimensional shapes. The image is a slice of a Fano variety, an example of a shape building block. (The slices inherit the curvature of the varieties.) The researchers hope their theory will ultimately help scientists in fields as diverse as computer vision, theoretical physics, and robotics. More images, animations, and mathematics can be found on the pair's blog.

Image: Slice of the Fano variety V6, by Andrew MacPherson and the FanoSearch team.

It was mathematicians to the rescue at an annual Maths-in-Industry Study Group, as reported in the Melbourne (Australia) daily The Age. Businesses from Australia and New Zealand paid $7000 AUD a head to have a group of 80 leading mathematicians from the public and private sectors bring a fresh perspective to some of their pressing--and costly--problems. Past problems include: How many apples can be packed in a box? Why are train carriages in the Adelaide Hills squeaky? How can washing machines be kept from shaking during the spin cycle? This year, mathematicians sought ways to monitor water quality and secure wind farm power systems. Hosted by the Royal Melbourne Institute of Technology, the annual event has brought mathematics to bear on 88 diverse conundrums since it was begun in 1993. The results are rewarding both for mathematicians, who appreciate the intellectual challenge, and for the participating companies, who one participating statistician estimates save millions of dollars through the Study Group. As the article quotes Connal Holmes of NZ Steel, "We find some quite innovative solutions coming through this group... these guys are not necessarily experts in our field but they come up with some amazing ideas from left field. We have some incredible solutions which literally we will be putting into practice next week."

A series of 90-second video spots, called Discoveries & Breakthroughs Inside Science, airs on local television stations around the country. These news segments are produced by the American Institute of Physics and supported by a group of scientific and engineering professional societies, including the AMS. The goal of this series is to promote awareness of and appreciation for the impact of recent advances in math, science, and engineering. In this episode, Dr. Uwe Bergmann, a physicist at SLAC National Accelerator Laboratory, describes how he used the Stanford Synchrotron Radiation Lightsource to x-ray the pages of an over 700-year-old prayerbook…and discovered the writings of Archimedes. In particular, researchers discovered a new word that "changes completely Archimedes's interpretation of the concept of infinity." These very powerful x-rays have also been used to provide new information about a 150 million year-old fossil, "considered one of the most important transitional fossils between dinosaurs and birds," Uwe notes. Similar techniques are being used elsewhere to uncover a painting hidden below a Van Gough painting and a musical score, composed by Luigi Cherubini, which was previously covered by carbon black ink. Watch the video.

Looking for a few fast facts about the concept of "zero" and its origins? This short article has them, starting with the pre-zero days of Greece and Rome. As the author notes, the zero--now a unique and basic part of mathematical learning--was only recognized as a "number" like all the others in the past several hundred years. The Babylonian "zero" was simply a gap in the writing on a tablet, and the Incan "zero" was an un-knotted segment within a sequence of knots on a string used to represent numbers. The first documented use of a symbol to represent zero is at an Indian temple built in 876 CE, and the word "zero" has its origins in Italian and Arabic.

In this episode of Making Stuff: Smarter, Nova's David Pouge visits mathematician David Smith at Duke University in search of an invisibility cloak. In collaboration with another mathematician, Sir John Pendry, David has developed a circular plastic encasing with tiny copper rings that effectively lets light-waves pass around the circle, making what ever is within invisible. Currently the main drawback of this invisibility cloak is that it only works for wavelengths in the microwave region and not light-waves visible to the human eye.

Photograph of FR4 cloak (by David Schurig). Read more and see additional images on the Novel Electromagnetic Media, the research group of David R. Smith at Duke University.

Estimating the number of people in a large crowd, such as a street protest, is harder than one might think. Due to Hosni Mubarak stepping down as president of Egypt after massive protests in Tahrir Square, the Wall Street Journal reports on the methods that journalists use to estimate crowd sizes. Even though the aerial photographs and satellite images have increased in quality, current methods used to determine the size of a large crowd are often no better than educated guesses and estimates can vary widely. Mathematicians and computer scientists are currently working on better techniques to estimate crowd sizes in accurate and unbiased manner.

Canadian mathematics professor James Stewart lives in "one of North America's most important private houses," according to the director of the Museum of Modern Art in New York City. Stewart's five-story "Infinity House," which is perched on the edge of a ravine in Toronto and is estimated to have cost $30 million, contains a 150-seat performance space and is a sought-after benefit locale for community interests. The secret to his success? Calculus textbooks. Ninety percent of Canadian university students and seventy percent of American university students learn from Stewart's chapters, and royalties earned on his book since it became a bestseller in 1992 have enabled Stewart to live his architectural dreams (although the author's conjecture about the value of the house is wildly overestimated). He began writing the text in the 1980's at the urging of his students, and it took him seven years to complete it. Read more about the house in a 2009 article in Focus, "James Stewart and the House That Math Built." (Photo by Ed Burtynsky.)